During my career I’ve seen great changes in the presentation of lectures and the production of handouts.

The original lecture technology, writing on a blackboard with a piece of chalk, is hopelessly old-fashioned now, though I still prefer it for teaching. It was replaced first by whiteboards, whose pens originally used foul-smelling chemicals (they have improved but I am not convinced they are healthier than chalk). Then came overhead projectors. Originally these tried to emulate boards: the acetate was on a roll, and you could turn the handle as you went and write continuous text, and could scroll back (the word seems more fitting for this than for a computer screen). Now OHPs are a health hazard: not, as you might think, because you might injure your back picking them up from the floor, or because you might trip over the lead, but because you might walk through the beam and damage your eyes.

Currently, computer and data projector rule, until they are superseded by the next thing, which might be some sort of smart paper. Mathematicians like the current set-up because the Beamer package gives us access to all the facilities and power of LaTeX.

What about the production of lecture notes or handouts?

I’ve always regarded lecture notes as not identical to what I write on the board or display on the screen, and write them with some care. But this effort, while fine for teaching, is not always appropriate. For seminars or conference presentations, a copy of the slides might be better.

Technology has changed things here as well. Once, lecture notes were handwritten or typed onto stencils, and duplicated by machines using even more unpleasant chemicals. Then came the photocopier, which took the pain out of duplication; high-quality laser printers and LaTeX did the same for preparation.

But reproducing slides has always posed problems of its own. By their nature, they are large format, one slide to a page (or many, if you use Beamer, since each \pause command starts a new page in the PDF). There was a program called mpage which took your PostScript slides and put them four (or two or eight) to a page. To produce PostScript from LaTeX, the standard route was via DVI. But now I only use pdfLaTeX, so this is not immediately viable. You can convert PDF to PostScript by the “print to file” command, and re-convert to PDF using ps2pdf.

The gap was filled, for a time, by various packages (my colleague Peter Kropholler produced one) which took the input file for the sldes and printed it out as a continuous document. Indeed, there is a package called beamerarticle which does this for Beamer input. This is how I produced the notes for my previous LTCC intensive course on Synchronization, which you can find here.

But recently I have found that there is a way to print the Beamer slides as a handout; I shall probably use this method for this month’s LTCC intensive on Laplacian eigenvalues and optimality. This may be useful to others, so here are the details.

The first step is to persuade Beamer to print one frame to a page, ignoring the \pause command and its refinements. This is easily done with the option handout. So if you compile your document with first line

\documentclass[12pt,handout]{beamer}

you will have taken this step.

To put several slides on a page, the versatile pdfpages package is what you need. Make a LaTeX input file using this package which has a single line between \begin{document} and \end{document}, reading

\includepdf[<options>]{file.pdf}

where file.pdf is the output from the previous step. Options include page layout (landscape or not, number of slides across and down, spacing, whether they are framed). I recommend you to this page; I have used it as is, except for putting six rather than four slides on a page.

Only one small problem remains. If you called the file that invokes pdfpages something generic like handout.tex, then of course the output will be called handout.pdf. I would prefer to have a name based on that of the input PDF. Probably someone has already done this; but when I am not so busy (maybe when I retire) I might try to produce my own.

15 Responses to Beamer handouts

Ah yes – Slides. I remember the days of photographing large typeface black type on white paper to produce 35mm black and white high contrast lith negs, then processing the negs with a colour forming developer to produce a blue background with white lettering for projection…!

The presentation used two 35mm optical slide projectors (Europe / USA beamers) with a crossfade dissolve system (I still have the system!)

I was looking at your Synchronization notes. I have a very elementary question that keeps bothering me in this context.

Using standard set notation, you introduce Ω as the set of integers from 1 through n. Then, “The full transformation monoid Tn is the set of all functions from Ω to itself.”

But — and I am not trying to be pedantic, this genuinely confuses me — a set is by its nature unordered. We apply the familiar bijective mapping, but it does not formally alter the set. Shouldn’t the underlying object be called a sequence, rather than a set? Otherwise we will never be able to observe anything about the operations which have been performed on it.

If the integers were identical red balls, your objection would be valid. But they all have different names, even if these names don’t make an ordering. If I have a bench with an apple, orange and banana on it, then you can tell if I swap the apple with the banana.

If I can tell the difference in the set after you have swapped two elements, it seems to me I must be perceiving some sort of position or slot which is (temporarily) mapped to each element.

If the elements did not have positions, they would all appear to be translucent and superimposed on each other. I could see that there was a banana and an apple, but that would be all. Inclusion and identity of the elements would be all I could learn by inspecting the set.

But if the elements were associated with distinguishable slots, then I could give the set to someone and thereby transmit more information, in addition to membership and the identity of each element. That seems wrong.

On the other hand, if the elements of the set do not thereby acquire positions or any other extrinsic attributes, I can point out the set to an observer, then permute the elements, indicate the same set again, and the observer will not be able to tell that I did anything. Even I will not be able to tell that I did anything.

Maybe there should be a postulate to the effect that interchange is detectable. Another (maybe better) possibility is that reference to an underlying set is superfluous, and that only the bijections have significance. The set, like the now-discarded aether of physics, vanishes from the theory, leaving only the rearrangements behind, like waves which do not, after all, require a medium in which to travel.

I realize that common a common sense argument about knowledge and experience practically guarantees I am wrong about all this, even if I can’t see why. Even as a child, I always had an unfortunate habit of asking many — possibly too many — questions. Over time I have acquired a great deal of knowledge and information by continuing to do so. But I know my questions can be annoying and argumentative. If that is the case now, I apologize.

Perhaps what I said was a bit misleading. Let’s be thoroughly mathematical about it. Consider the set {a,b,c} (if you prefer, an apple, a banana and a cabbage). As you correctly say, {c,b,a} is the same set, so how can the permutation interchanging a and c be defined?

I build an auxiliary structure from this set, namely, the set of ordered pairs. Choose two elements (the same or different) from the original set, and give them distinguished slots “first” and “second”. Collect these into a set with nine elements.

Now choose three of the nine elements in this set with the properties that each of a,b,c occurs first in one of the three, and each occurs second in one of the three. Call such a collection a permutation. This is a passive way of thinking about the function which maps the first element of each pair to the second; being passive, it perhaps avoids the problems you raise: it is just a set, nobody does anything.

So, for example, {(a,c),(b,b),(c,a)} is the permutation interchanging a and c.

There is one further subtlety. As you say, a, b and c come in no particular order (if we have forgotten our ABC), so we could number them 1, 2 and 3 arbitrarily. If you do that, you will find that there are three different permutations of {1,2,3} that you get, namely the three transpositions. They form a conjugacy class in the symmetric group of degree 3. This is true in general; if you do this to any permutation of an n-element set (as defined above), you obtain a well-defined conjugacy class of the symmetric group.

In a sense, conjugacy is the relation on permutations that you get if you don’t regard the things they act on as being ordered.

The first post in my series about the symmetric group is vaguely relevant here.

(More than four years later, I am still pondering this exchange with you. My mind, which has always been slow at mastering really new concepts, is now, at age 65, glacial. But on to the point at hand…)

Since I asked my question, I’ve come to accept, reluctantly at first, that the most straightforward way to define a permutation is as a bijection. But I did not understand that at the time I wrote to you.

Thus now I want to ask, was there a reason you did not employ the concept of a bijective function in your (kind) reply to my question? Although (x, f(x)) is an ordered pair, an element of the cartesian product of X and F(X), it seems unnecessary to describe it as such when using the definition based on a bijection, because the concept of a function contains a built-in arrow, namely from domain to codomain.

There are various ways of thinking about a permutation. To some people it is a set of points in a square grid, with just one in each row and column; the permutation takes the column index to the row index of the point. The coordinates of the points give the representation of the permutation by ordered pairs.

My preferred way of regarding a function is a black box which takes an input and produces an output; the black box comes with a manual which tells you what elements are permitted as inputs (the domain of the function) and also gives you a set in which the output is guaranteed to lie (the codomain). To be a permutation, the manual must guarantee that the domain and codomain are the same, and that every element of the codomain is the output from exactly one input.

In my example with three pieces of fruit, the black box is a robot, I guess …

I like your explanation using ordered pairs. I only have to know how to count to 2! And that exclamation point is not even ambiguous!

One might object that the usual way of explaining “permutation” disturbs only me, but surely the method you have just presented is better all around.

Your (is it new?) exposition might even help clear up the awful problem of notation — I mean how to write down a specific permutation, for practical use. One should never try to explain how to write down a permutation by rearranging {1, 2, 3…}. It only leads to heartache, just another consequence of the same “underlying set” confusion — if you ask me — which of course no one ever does.

“a, b and c come in no particular order (if we have forgotten our ABC), so we could number them 1, 2 and 3 arbitrarily. If you do that, you will find that there are three different permutations of {1,2,3} that you get, namely the three transpositions.”

Three different permutations that you get, sorry, how? Why not the usual six permutations?

It is only three. If you choose to number a, b, c with 1, 2, 3, you get the permutation 1->3, 2->2, 3->1 (this is shorthand for the set {(1,3), (2,2), (3,1)} of ordered pairs). If you number them differently, say 2, 3, 1, then you get 2->1, 3->3, 1->2, which is a different set of pairs. But just three different sets will arise, not all six possibilities.

If you take the permutation {(a,b), (b,c), (c,a)} and number a, b, c with 1, 2, 3 arbitrarily, you can get either 1->2->3->1 or 1->3->2->1.

The classes that arise are precisely the conjugacy classes in the symmetric group.

Moving to St Andrews has given me the opportunity of making a new start. From now, when I put slides of talks on the web, I will use this code, and I have put the code on the web page at http://www-circa.mcs.st-andrews.ac.uk/~pjc/talks/ in case I forget it.

There has been some discussion recently about the fact that Beamer handouts no longer work as claimed. It seemed, after the discussion, that the problem is with pgfpages rather than with beamer. Various fixes have been suggested, and there is a claim that recent versions are OK. My code referenced above uses pdfpages. I have never noticed any problem with it.